Question

Use a computer algebra system to graph the vector-valued function and identify the common curve. $$ \mathbf{r}(t)=-\sqrt{2} \sin t \mathbf{i}+2 \cos t \mathbf{j}+\sqrt{2} \sin t \mathbf{k} $$

Short answer

The graph of the given vector-valued function forms a helix around the y-axis with the help of a Computer Algebra System.

Step-by-step solution

Identify components

First, separate the vector-valued function \(\mathbf{r}(t)\) into its x, y, and z components. In this case, these components are: \(x(t) = -\sqrt{2} \sin t\), \(y(t) = 2 \cos t\), and \(z(t) = \sqrt{2} \sin t\).

Enter function into a computer algebra system

Using a computer algebra system like Mathematica, Matlab, or an online graphing tool suitable for three dimensions, enter each of the components from Step 1 as a function of \(t\). Typically, this involves specifying each coordinate of the function as a separate expression.

Graph the vector-valued function

Graph the function using the graphing capabilities of the computer algebra system. Make sure to properly set the limit for the variable \(t\) to get the most accurate graph.

Identify the common curve

Once the graph has been generated, observe and identify the common curve in the graph. The shape will depend on the specific function, but it could be a line, plane, spiral, or any number of other shapes. In this case, due to the presence of sine and cosine functions, it is expected to be a helix around the y-axis.